What criteria do I use when choosing which APIs to download? [closed]

It depends on which audience your app is targeting and the level of compatibility you want.

You can download the latest summer, but if you want your app to be compatible with older versions of Android, you should set the minimum version.

Below, you are set to use SDK 21 (Lollipop), but maintaining support for SDK 14 (Android 4.0)

defaultConfig {
    applicationId "com.foo.meuapp"
    minSdkVersion 14
    targetSdkVersion 21
}

Or AndroidManifest.xml:

<uses-sdk android:minSdkVersion="14"
          android:targetSdkVersion="21" />

To support 90% of Androids you should now develop your app with minimal support for API 14 ( reference ).

You have to download the ones you need, I recommend downloading the 2.3.3

And all of 4.4 up.

Do not download the "EXAMPLES", at least I do not consider it necessary.

In short, download whatever you find necessary.